Euler tours and unicycles in the rotor-router model

TL;DR

This paper investigates the properties of rotor-router walks on finite graphs, focusing on the structure of recurrent states, the sequence of cycles generated, and the walker's displacement, revealing both probabilistic and invariant features.

Contribution

It introduces a detailed analysis of unicycles and cycle sequences in rotor-router models, including expected cycle counts and invariants related to planarity.

Findings

Expected numbers of dimers and contours calculated

The difference between contours and dimers is an invariant

Mean-square displacement analyzed in recurrent states

Abstract

A recurrent state of the rotor-routing process on a finite sink-free graph can be represented by a unicycle that is a connected spanning subgraph containing a unique directed cycle. We distinguish between short cycles of length 2 called "dimers" and longer ones called "contours". Then the rotor-router walk performing an Euler tour on the graph generates a sequence of dimers and contours which exhibits both random and regular properties. Imposing initial conditions randomly chosen from the uniform distribution we calculate expected numbers of dimers and contours and correlation between them at two successive moments of time in the sequence. On the other hand, we prove that the excess of the number of contours over dimers is an invariant depending on planarity of the subgraph but not on initial conditions. In addition, we analyze the mean-square displacement of the rotor-router walker in the recurrent state.